The five "normal" girls also have male rivals who are trying to win their hearts. Each of these rivals have a set of Rival Events just like your own Heart events. If you see all 4 of the Rival Events then the couple will marry, whether you want them to or not.
The girls from Mineral Town and the other special girls do not have any rivals to contend with. Daryll will not try and stop you if you plan on marrying his ward, Leia the mermaid. Cliff, Gray, Kai, Doctor, and Rick do make appearances in your game if you connect to FoMT but they won't try and win away Ann, Mary, Popuri, Elli, or Karen.
The Rival Events will occur if a girl's heart color is at the matching rival heart requirement level or below. That is opposite of your own Heart Events, which can occur if the girl is at the specified color or above. You can activate a Rival Event at black, blue, green, and orange heart colors. The events also must go in order so if you haven't seen the blue Rival Event and the girl is at a green color, the event will not activate until she is back to a blue color and you can't skip ahead to see the orange event.
The orange rival event will not trigger until you have rescued the required 60 Harvest Sprites necessary for marriage.
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The Black Rival event can happen at the beginning of the game and beyond, but the Blue Rival event can only occur if you've reached your 2nd year. Likewise you need to be on your 3rd year or later for the Green Rival event to happen, and the last Orange event won't happen until you have reached your 4th year. That gives you 4 years to woo any particular girl without having to worry about triggering the last rival event.
After you see the last rival event the couple will get married 1 week later. The boy will come by your house in the morning and invite you to the ceremony. You don't get any chance to object to the wedding and your entire day is spent at the Beach. That means your animals and crops will not be cared for that day. Not even the Sprites will do any work!